Numerical simulation and optimization of SiC

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Numerical simulation and optimization of SiC single crystal growth by sublimation from the gas phase

Collaborator: O. Klein, P. Philip, J. Schefter, J. Sprekels

Cooperation with: Ch. Meyer, A. Rösch, F. Tröltzsch (Technische Universität Berlin), K. Böttcher, D. Schulz, D. Siche (Institut für Kristallzüchtung (IKZ), Berlin)

Supported by: BMBF: ``Numerische Simulation und Optimierung der Züchtung von SiC-Einkristallen durch Sublimation aus der Gasphase'' (Numerical simulation and optimization of SiC single crystal growth by sublimation from the gas phase); DFG-Forschungszentrum ``Mathematik für Schlüsseltechnologien'' (Research Center ``Mathematics for Key Technologies''), project C9

Description: $% latex2html id marker 38471 \minipage{0.55\textwidth}\Projektbild {\textwidth}... ...owth apparatus according to [\ref{lit1_sic_Pon}]\label{fig1_sic_1}} \endminipage$ Owing to numerous technical applications in electronic and optoelectronic devices, the industrial demand for high quality silicon carbide (SiC) bulk single crystals remains large. It is still a challenging problem to grow sufficiently large SiC crystals with a low defect rate. Sublimation growth of SiC bulk single crystals via physical vapor transport (PVT), also known as the modified Lely method, has been one of the most successful and most widely used growth techniques of recent years.

During PVT, a graphite crucible (see Figure 1) is placed in a low-pressure inert gas atmosphere consisting of argon. The crucible is then intensely heated, e.g., by induction heating , to temperatures up to 3000 K. Inside the crucible, polycrystalline SiC source powder sublimates, and the gaseous species diffuse through the cavity from the powder to the SiC seed. As the single-crystalline seed is kept at a temperature below that of the SiC source, the species crystallize on the seed, which thereby grows into the reactor.

The physical and mathematical modeling of the growth process leads to a highly nonlinear system of coupled partial differential equations. In addition to the kinetics of a rare gas mixture at high temperatures, one has to consider heat transport by conduction and radiation, reactive matter transport through porous and granular media, and different kinds of chemical reactions and phase transitions. The main control parameters with respect to an optimization of the crystal growth process are the design of the growth apparatus, the position of the induction coil, the heating power, and the inert gas pressure.

The heat sources caused by induction heating are computed via an axisymmetric complex-valued magnetic scalar potential that is determined as the solution of an elliptic PDE using the imposed voltage as input data. The scalar potential enables one to calculate the resulting current density and thus the heat sources (see [1, 2] and the references therein).

Within the covered research period, the simulation software WIAS-HiTNIHS has been tested at the IKZ by comparing, for a simple version of the equations, the results derived with WIAS-HiTNIHS with results computed by other software. This has led to an enlargement of the region considered for computations and to some improvements of WIAS-HiTNIHS.

In [4], based on the numerical solution of the stationary mathematical model for the heat transport in the growth system by WIAS-HiTNIHS, a Nelder-Mead method has been used for a numerical optimization of the control parameters frequency f, power P, and coil position z_rim for the radio frequency (RF) induction heating of the growth apparatus. The control parameters have been determined to minimize a cost functional, being either the L-norm of the radial temperature gradient on the single crystal surface or the L²-norm $\mathcal {F}$ _{r, 2} of this radial gradient, or $\mathcal {F}$ _{r, 2} minus the L²-norm $\mathcal {F}$ _{z, 2} of the vertical temperature gradient between SiC source and seed. The optimizations have been subject to constraints with respect to a required temperature difference between source and seed, a required temperature range at the seed, and an upper bound for the temperature in the entire apparatus.

Several series of Nelder-Mead optimizations of (P, z_rim) have been performed, varying the used initial values and keeping the frequency f fixed. Moreover, also the functional dependence of the cost functional on (P, z_rim) as well as the restrictions imposed by the state constraints have been studied, varying the power P and the coil position z_rim and performing, for each (P, z_rim), a forward computation to compute temperature fields T = T(f, P, z_rim) and the corresponding value of the cost functionals.

**Fig. 2:** Contour plots of the functionals $\mathcal {F}$ _{r, 2} (left-hand side) and ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{r, 2} - ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{z, 2} (right-hand side) restricted to the part of the (P, zrim)-plane where the state constraints are satisfied. Therein, a star marks the location where the smallest value of the considered objective functional occurred during the series of forward computations discussed, and the locations of results of 9 corresponding 2-dimensional Nelder-Mead computations (keeping f = 10 kHz fixed) are indicated by dots, which can be found on the lower edge of the respective admissible region.
$\makeatletter \@ZweiProjektbilderNocap[h]{0.46\textwidth}{fb03_1_02_sic_fig2.eps.gz}{fb03_1_02_sic_fig3.eps.gz} \makeatother$

Moreover, we have performed three-dimensional Nelder-Mead optimizations, controlling f in addition to P and z_rim. As in the two-dimensional optimizations, different objective functionals have been considered. Varying the initial values, series of 27 Nelder-Mead computations have been performed. The results for two series are shown in Figure 3.

**Fig. 3:** Result values for (P, zrim, f ) for two series of three-dimensional Nelder-Mead computations, with a color presentation of the corresponding values of the objective function $\mathcal {F}$ _{r, 2} (left-hand side) and ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{r, 2} - ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{z, 2} (right-hand side). The big point in each figure corresponds to the triple with the respective lowest value of the considered objective function.
$\makeatletter \@ZweiProjektbilderNocap[h]{0.45\textwidth}{fb03_1_02_sic_fig4.eps.gz}{fb03_1_02_sic_fig5.eps.gz} \makeatother$

The effect of the respective minimizations of $\mathcal {F}$ _{r, 2}(T) and of ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{r, 2}(T) - ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{z, 2}(T) on the shape of the temperature distribution between SiC source and crystal is portrayed in Figure 3. Therein, one has the stationary solution for a generic, unoptimized situation, and also the solutions with the lowest values for $\mathcal {F}$ _{r, 2}(T) and for ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{r, 2}(T) - ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{z, 2}(T) found within the three-dimensional Nelder-Mead computations, which are marked in Figure 3.

As a result of the optimizations, a minimal radial temperature gradient is found to coincide with a minimal temperature at the single crystal surface, and a maximal temperature gradient between source and seed is found to coincide with a low coil position.

In [5], a quite general version of transient nonlinear and nonlocal heat transport equations has been discretized using an implicit Euler scheme in time and a finite volume method in space. For the corresponding nonlinear and nonlocal discrete scheme, the existence of a unique discrete solution has been proved and discrete L-L¹ a priori estimates have been established. In [3], a less general version of the equations has been considered, and the existence of a unique discrete solution to the corresponding discrete scheme has been proved under weaker assumptions for the discretization, and, moreover, a discrete L-L a priori estimate has been derived.

**Fig. 4:** The shape of the temperature distribution between SiC source and crystal. The distribution on the top corresponds to a generic, unoptimized situation, using f = 10 kHz, P = 10 kW, and zrim = 24 cm. The lower figures present temperature distributions corresponding to those values for (f, p, zrim) with the lowest values for $\mathcal {F}$ _{r, 2}(T) (left-hand side) and of ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{r, 2}(T) - ${\frac{{1}}{{2}}}$ $\mathcal {F}$ _{z, 2}(T) (right-hand side) found by the three-dimensional Nelder-Mead computations, which have been marked in Figure 3.
$\makeatletter \@DreiProjektbilderNocap[d]{0.49\textwidth}{fb03_1_02_sic_fig6.eps.gz}{fb03_1_02_sic_fig7.eps.gz}{fb03_1_02_sic_fig8.eps.gz} \makeatother$

References:

O. KLEIN, P. PHILIP, Transient numerical investigation of induction heating during sublimation growth of silicon carbide single crystals,
J. Crystal Growth, 247 (2002), pp. 219-235.
, Transient temperature phenomena during sublimation growth of silicon carbide single crystals, J. Crystal Growth, 249 (2003), pp. 514-522.
, Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme, WIAS Preprint no. 871, 2003.
C. MEYER, P. PHILIP, Optimizing the temperature profile during sublimation growth of SiC single crystals: Control of heating power, frequency, and coil position, WIAS Preprint no. 895, 2003.
P. PHILIP, Transient numerical simulation of sublimation growth of SiC bulk single crystals. Modeling, finite volume method, results, Ph.D. thesis, Humboldt-Universität zu Berlin, 2003, WIAS Report no. 22, 2003.
M. PONS, M. ANIKIN, K. CHOUROU, J.M. DEDULLE, R. MADAR, E. BLANQUET, A. PISCH, C. BERNARD, P. GROSSE, C. FAURE, G. BASSET, Y. GRANGE, State of the art in the modelling of SiC sublimation growth,
Materials Science and Engineering B, 61-62 (1999), pp. 18-28.